# Algebraic Structures Matrix

• May 18th 2008, 09:06 AM
mrks1d
Algebraic Structures Matrix
I really need help, i need to investigate the properties of the set of matrices
a -b
b a :a,b belongs to real numbers

I need to know
• transformation properties
• diagonalisation
• group properties
• subgroups
• Isomorphism
Any help would be fantastic!

thanks!
• May 18th 2008, 09:21 AM
Isomorphism
You will have to be more specific than that to get useful answers. For the second part, I dont think you can diagonalize it(over reals) since its eigenvalues are always complex.

What transformational property do you want? What group, subgroup properties are you looking at?
• May 18th 2008, 11:37 AM
mrks1d
I think that is why i am having the problem! That is exactly how the question is worded no extra information is given. (Worried)
• May 18th 2008, 11:44 AM
Moo
Hello,

Quote:

Originally Posted by mrks1d
I think that is why i am having the problem! That is exactly how the question is worded no extra information is given. (Worried)

But it's not a problem to diagonalise it over $\displaystyle \mathbb{C}$ o.O
The eigenvectors will remain over $\displaystyle \mathbb{R}$

$\displaystyle \chi_A(\lambda)=(a-\lambda)^2+b^2=(a-\lambda-ib)(a-\lambda+ib)$

$\displaystyle \lambda_1=a+ib$
$\displaystyle \lambda_2=a-ib$

Then find the eigenvectors. To solve for X(x,y), representing the eigenvector, just remember that if a complex number is null, it means that its real part is zero and its imaginary part is zero..